The hypothesis of the ether was introduced to account for the transmission of light, heat, electricity, and so forth, and has proved of the utmost service to physicists. Most of my readers are probably acquainted with the general idea and I need not therefore discuss it in detail.
It will be sufficient here to say that it is supposed to be a weightless, homogeneous medium extending throughout all space and permeating all bodies. Indeed Matter itself is supposed to be no more than the result of more or less complex disturbances in it.
But although it accounts for the phenomena in connection with which it was called into being it is necessary to ascribe to it very contradictory properties. On the one hand it has been calculated that in order for it to transmit the forces which we know that it does transmit, for instance the force of gravitation, it must possess a rigidity some 3,000 times greater than that of the strongest known steel. On the other hand we must suppose it to be of a tenuity far in excess of the most perfect vacuum which we can obtain, for otherwise the earth and other planets which are moving at immense speed through this medium would be slowed down; which is not in practice the case.
Now Hinton points out that to a two-dimensional being, a perfectly smooth solid sheet on the surface of which he lived would possess many of these properties. Being perfectly smooth it would be imperceptible to him and would offer no opposition to the passage of bodies over it. Yet it could, being solid, transmit vibration just as we know the ether does for us. Also it could be as rigid as you please without losing any of its imperceptibility. It could not be weighed and it could not be eliminated from any vessel no matter what care was taken to do so.
The analogy is striking but it does not appeal to me and I do not think that even Mr. Hinton means it to be taken strictly, for in other passages he gives quite different suggestions as to the ether.
One of the latter is derived from a consideration of the phenomena of rotation in four-dimensional space and is of some intrinsic interest.
In two space rotation takes place about a point, in three space about a line and we should therefore expect that in four space it would do so about a plane. This is easily shown to be the case although I do not propose to go into the proof here. The only important point is that whereas it is impossible to conceive a mass of three-dimensional spheres in a state of continuous rotation,—because they would be trying to drive each other in different directions and so would prevent the rotation,—in four dimensions this is not the case and a mass of "hyper-spheres" could be "self-driving," that is to say the rotation of each could be such as to assist and not to retard that of its neighbours. This fact is of interest because Lord Kelvin showed that the contradictory properties of the ether referred to above could only be reconciled by supposing it to be animated throughout by a motion of a vortical character.
This "self-driving" effect of rotating hyper-spheres is worth glancing at a little more closely. It arises from the fact that there are two distinct sorts of rotation which such a sphere may possess. In three-dimensional rotation the motion may take place about any axis we please and the other two axes which can be drawn will change one into the other, so to speak, as the rotation takes place. But in four-dimensional space we have four axes and while the X and Y axes change place, say, there is nothing to prevent the W and Z axes doing so too. Thus we might have the X axis changing into the Y and the W into the Z. To reverse both of these motions so as to have the Y axis changing into the X and the Z into the W does not give us a new kind of motion any more than reversing the direction of an ordinary three-dimensional rotation does—it is only equivalent to looking at it from a different point of view. But if in the case of the four-dimensional rotation we reverse one only of the two rotational components we do get a new kind of motion, and this is of interest in view of the fact that electricity like other forces is regarded as a mode of etheric motion, and if this be so there would seem to be a certain need for two distinct kinds of it in order to correspond to positive and negative electricity respectively.
It is just possible that there is some connection, as Mr. Hinton suggests, between this need and the two kinds of four-dimensional rotation referred to above.